Furnishings are divided up into 6 unnamed categories. There is a cost to installing any furnishing, and many furnishings have a seasonal upkeep cost. However, they may raise the rent and comfort stats of the apartment, allowing you to charge more for your apartments, get larger deposits, and attract wealthier clients.
To unlock most furnishings you need to activate some trigger event, like having a resident successfully enter a certain profession or winning the item in a scratch card. After this, you can research the furnishing to unlock the ability to build that furnishing in your apartments. For most items, researching once gives you access a limitless quantity of the furnishing. However, there are some exceptions where you need to researching the item any time you want another copy; these are noted with a superscript infinity symbol after the name (∞).
Some furniture can only be obtained from Scratch Cards or Gold Cards. Scratch Cards cost 4 Tickets and allow you to scratch off 1 item each (which may or may not be furniture). Gold Cards cost 10 Tickets and allow you to scratch off 3 items (which may or may not be furniture). Note that Scratch Cards and Gold Cards give you access to different items.
Exclusive to Scratch Cards: Low Table, Floor Cushion, Study Desk, Wood Chair
Most furnishings have a level attribute. They start at level 1 and, as they are used by tenants, increase to a max level of 10. The +Rent and +Comf(ort) reported in tables is the stat mod given when the furnishing is at level 10. (NB: Furnishings that cannot be leveled have a superscript dash affixed to their name.)
You can use items on your furnishings to raise their stats. The Comfort Booster and Rent Booster are obvious choices, but any item whose description begins with "Boosts..." will improve your furniture in some way.
Category 1 type furnishings consists primarily of flooring, but it's also where you find the Interior Door, rearrange, and remove options. These are the simplest furnishings in the game.
Balcony flooring is also available, although it is unknown if you can earn and build this. You can get this flooring when you start a game by choosing the city Bigville or by expanding with Big Apartment 3.
Category 5 furnishings include rugs, wall fixtures, lawn ornaments, and plants. All furnishings in this category have special placement rules. Additionally, with few exceptions (Plant, Flower Bed), Category 5 items must be researched each time you want to increase your inventory.